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A Hilbert space approach to maximum entropy reconstruction
Author(s) -
Klaus Martin,
Smith Robert T.
Publication year - 1988
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670100405
Subject(s) - mathematics , hilbert space , piecewise , optimization problem , residual , entropy (arrow of time) , mathematical optimization , attenuation , mathematical analysis , algorithm , physics , quantum mechanics , optics
We examine here the problem of reconstructing an X‐ray attenuation function from measurements of its integrals. The approach that is taken is to maximize the difference of the entropy and the residual error in meeting the measurements. The solution of this optimization problem is constrained by requiring that the solution lie in a certain weakly compact subset of L 2 , to be determined by physical information. We show that the constrained optimization problem is well‐posed: there exists a unique solution (even when the measured data are inconsistent) and the solution depends continuously on the measurements. In the course of proving this, we show that the entropy functional is continuous on L 2 . We further demonstrate that the solution of the optimization problem for a special case, must be piecewise constant.